Sunday, February 19, 2012

Before we get started, I'd like to thank listeners Foxy McLovin and The Devonian Kid, who recently posted nice reviews on iTunes. Remember, you too can experience the thrill of having your name, or bizarre iTunes nickname, mentioned on Math Mutation by posting a review. Or you can send a donation to your favorite charity in honor of Math Mutation, and email me to tell me about it. Anyway, on to today's topic.

During this Valentine's week, it occurred to me that there was not yet a Math Mutation episode focusing on amicable numbers, said to be numbers that represent friendship and love. Yes, there are such things. And yes, math geeks do occasionally experience such emotions. So what exactly are amicable numbers?

To start with, let's review the concept of a perfect number. A perfect number, like 6 or 28, is precisely equal to the sum of its factors. So 6, for example, is perfect because its factors are 1, 2, and 3, and 1 + 2 + 3 = 6. The concept of amicable numbers is related to perfect numbers, except that they are pairs of numbers, such that each is the sum of the other's factors. The smallest pair of amicable numbers is 220 and 284. The factors of 220 are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55 and 110, which add up to 284. And the factors of 284 are 1, 2, 4, 71 and 142, which add up to 220. If you stretch your sense of analogy a but, you can kind of see how the way in which two numbers are unexpectedly intertwined might suggest love and friendship.

The pair 220 and 284 was known to the ancient Greeks. But as you can probably guess, amicable numbers are relatively rare, and can be tricky to find. In 850 A.D, Arabic mathematician Thabit ibn Kurrah discovered if n > 1 and each of p = 3*2^n-1-1, q = 3*2^n-1, and r = 9*2^2n-1-1 are prime, then (2^n)pq and (2^n)r are amicable numbers. This was a nice advance, but finding numbers that meet the preconditions of this formula wasn't very easy-- several centuries later, it led to discovery of the pairs 17,296 and 18,416, resulting from n=4, and 9,363,584 and 9,437,056, which you get from n=7. Euler later published a list of 64 new amicable pairs, but he actually made some mistakes, and the list had 2 bogus entries, which was not discovered until the 20th century. But most surprisingly, the second-lowest pair of amicable numbers, 1184 and 1210, was not discovered until 1866, by a 16-year old Italian boy named Nicolo Paganini. With the help of modern computers, millions of amicable pairs have been discovered, though the conjecture that there are an inifnite number of them has not yet been proven.

Amicable numbers have been thought to symbolize love and friendship since ancient times. It is said that when Pythagoras was asked to define a friend, his definition was "one who is the other I, such as are 220 and 284." In Genesis verse 32:14, Jacob gave his brother 220 sheep when he feared he was planning to kill him, and some Torah scholars interpret this as due to its status as an amicable number, to reinforce their brotherly love. I think this last interpretation is a bit of a stretch though-- he also gave him 30 camels, 40 cows + 10 bulls, and 30 donkeys, which don't seem to have anything to do with amicable numbers. And to get the number 284 to appear somehow, scholars have interpreted one partial word in the verse as a Hebrew encoding of a number, which really seems to be reaching.

According to one online site called "Renassance Astrology", you can make a magical love talisman by the following method: "Make two images, and put one of the Fortunes at the ascendant and the Moon in Taurus conjunct Venus. Write on one image a number ... for 220 in the proper place, and on the other image write the same kind of figure for 284 in the proper place. Join the two figures together in an embrace, and then there will be perfect and lasting love between the two." They go on to supply further mystical powers of these numbers: "If the aforesaid numbers were carved in wood, and bread or anything else edible was sealed with them and you gave it to someone to eat, he will delight in you with a great love. If those numbers are written on your clothing, your garments cannot be taken away from you; and if you write them on banners that are put in the street to draw business, they will draw business to you." I'm thinking of writing some of these numbers on my socks, to stop them from disappearing in the dryer.

And at least some modern geeks still believe that amicable numbers represent love: with a quick online search, I was able to find a pair of half-heart pendants, one labeled 220 and the other labeled 284, for sale at a site called mathsgear.com . But math geekiness can only get you so far in a marriage-- when it comes to Valentine's day, I made the safe choice and got my wife some non-math-related flowers. Perhaps you will be more courageous on your next romantic occasion.
And this has been your math mutation for today.

Note that this podcast is intended mainly for audio consumption, so you will not see the numerous illustrations & diagrams you would find at most math sites, though these are linked in the show notes whenever possible.

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